The following correction was printed in the Guardian's Corrections and Clarifications column, Wednesday June 8, 2005.

Simone Young, mentioned in the article below, is no longer music director of Opera Australia. She left that post in 2002-03.

One day, Marin Alsop boarded an aircraft and looked into the cockpit. There were three women inside, and there was every likelihood that they would be flying the thing. "My first reaction was 'Uh-oh'." Alsop is not proud of her instinctive response. "Of course, it turned out to be the smoothest flight I'd ever been on. But my reaction was very thought-provoking. I guess I'm as much a victim of societal programming as the next person."

This honest admission is especially surprising from Alsop, one of the very few women conductors in the world and one who is proud to call herself a feminist. What a horrible shock to find you had unconsciously internalised a sexist agenda. "Well, yes. When I watch TV news now, I notice that we've become accustomed to seeing a man and a woman presenting. Twenty years ago it was just all men. If it's two women it doesn't seem quite right." Why? "It's all about comfort levels. It's not all specifically about capability or connection of knowledge. It's much more abstract."

Conducting, one might well think, is fundamentally a sado-mascochistic ritual involving either virile young male turks whipping orchestras into frenzies with tapering batons of birch and maple, or sage old men waving their penis substitutes in front of a bunch of instrument-playing stiffs in irksome collars, before jetting to the other side of the world to do exactly the same thing in another cultural capital. Viewed thus, conducting is an unsuitable job for a woman, particularly one who, like this 48-year-old New Yorker, has childcare responsibilities.

Yet Alsop has made a name for herself in this man's world. The first CD in a projected Brahms symphony cycle that she released with the London Philharmonic earlier this year led the Guardian to pronounce that she was "well on her way to becoming one of the best Brahms interpreters". Last week, she won a Classical Brit (admittedly as best female artist) for her recordings of John Adams, Samuel Barber and Philip Glass. In 2003, she was voted Gramophone's artist of the year and also won the Royal Philharmonic Society's conductor award. She has been principal conductor of the Bournemouth Symphony, Britain's oldest orchestra, since 2002, reviving its artistic reputation after a disastrous period, and attempting to radicalise the tastes of what was regarded as a conservative audience with new music. She is conductor laureate of the Colorado Symphony and guest conducts for many of the world's leading orchestras. On Sunday, this protege of Leonard Bernstein will conduct the LSO in a performance of his Mass at the Barbican. It is a fine CV, even if the post of principal conductor at any of those top orchestras still eludes her.

But she has hardly any female peers. True, there is Simone Young, music director of Opera Australia. Yes, there is Andrea Quinn, music director of New York City Ballet. Ten years ago there used to be Sian Edwards at the English National Opera, until she abruptly left for "internal reasons". But very few women, living or dead, have taken to the podium. Back in the 1870s, there was apparently a woman conductor of the great Dutch orchestra, the Concertgebouw, about whom it would be interesting to know more. Next year, Alsop will guest conduct for that Amsterdam orchestra - more than a century after the last woman raised a baton there.

Does she not find this depressing? "I assumed 20 years ago when I got into the profession and looked around that there would be an influx of women on to the podium. It never happened." Yet, in that period, women have hardly been unobtrusive as instrumentalists. Think, for instance, of all the great women pianists - Martha Argerich, Mitsuko Uchida, Myra Hess, Moura Lympany and Annie Fischer. There is no such roll call of honour for women conductors.

The result, then, is that Alsop has to bear the burden of being both an exemplar to young women conductors and one who incessantly has to explain why there are few like her around. It is, one might think, a royal pain. "It's a question I am asked a lot." And what is the answer? Why are there so few women conductors? "It's a chicken and egg question. Maybe there aren't enough talented women conductors because there have never been many women conductors." She particularly finds one question on this subject especially daft. "When people ask me, 'What's it like to be a woman conductor?' I have no perspective. I've always been a woman conductor. I have never been, say, a lizard conductor."

Doesn't she get exasperated with having to answer the same questions? "I look forward to the time, a generation down the line, when things have changed so that it's no longer an issue." But will that happen? As she has previously said, rather optimistically, "When I teach it's generally a 50/50 ratio of men to women and I don't think women hesitate as much to consider a career as a conductor these days ... But it would be naive not to notice that there are no women music directors of any major orchestras in the world."

She was pleased when appointed as principal conductor of the Bournemouth Symphony Orchestra four years ago that the British media did not make much of the fact she was that rare thing, a baton-waving woman. "They would just have one sentence. It wasn't a huge deal." Why do you think that is? "Possibly because of Margaret Thatcher. It's very different in the US, especially in the classical music world, which is very conservative. It's a huge deal there to be a woman conductor. As a society we are not accustomed to seeing women in the highest offices."

When will Alsop rise to the highest offices? Some say that she took the Bournemouth job to build a British power base from which she would become principal conductor of one of London's leading orchestras. Last year she was tipped by the Los Angeles Times to become the principal conductor of the Los Angeles Philharmonic. She denies she has such a grand plan, pointing out that she has just renewed her Bournemouth commitment for two more years.

Alsop started learning the piano at two and the violin at five, but her real musical epiphany came when, aged nine, her father took her to see Bernstein conduct a Young Person's Concert in New York. "He was jumping around and talking to the audi ence. He was breaking all these rules. It was like he was in Technicolor, it was so captivating. I said: 'I'm going to do that' and then it took 30 years, but I did. I knew I wanted to do it and I never changed my mind." Her parents, Ruth and LaMar Alsop were professional musicians - she a cellist and he a violinist and concertmaster for the NYC Ballet orchestra. Were they encouraging? "Yes. There was absolutely no gender division in my family. My mother played the cello, for instance." She thinks for a second. "But dad used to carry it for her. They never thought it was a bad idea nor did I think about the woman issue until I had been conducting for a while."

But Alsop says she has reflected since on the gestures that she makes as a conductor in order to undermine her musicians' sexist prejudices. "The hardest thing for me is always to get a big sound from the orchestra without being very demanding or apologising. As a woman, if you're too aggressive people think, 'She's so overpowering. What's she on us for?'. But if a man does the same gesture, it's regarded as strong and virile." To show me what she means she sticks out an upturned fist towards me and then uncurls her fingers. "If you're too strong, the orchestra responds the same way. If you push them too hard, the sound tightens up. I have worked really hard to make my gestures less threatening. Also, when you're delicate, as a woman sometimes you're interpreted as being lightweight. Men don't get that."

At the choral rehearsal for the Bernstein Mass I attend at St Luke's, the LSO's London rehearsal space, her gestures do not seem to be misinterpreted. She is brisk but warm, charming to the small children in the choir and helpfully communicative to everybody else. She says she models herself somewhat on Simon Rattle, whom she regards as a great communicator (she is renowned for her pre-concert talks, where she is, like him, keen to contextualise new music for audiences often unfamiliar with such work).

This is the second rehearsal of the day. She flew in from Denver (nine hours to Heathrow) and went straight into rehearsal. Her hour-long break, punctuated by slugs of Diet Coke, was this interview. Now she is back putting the choir through the Mass's Kyrie Eleison.

She says she flies around the world seven times in a year. "Success equals hyperactivity in this industry. Jet transportation has made this life possible; conductors wouldn't choose it. It's particularly hard on women who have children. I have a son and I want to stay with him at home in Denver. So I try not to be away from home for more than two weeks." But there is a counterveiling impulse - her love for music. "I'm just trying to follow the passion for music. I try to do what feels thrilling in every dimension of my life." She realises this sounds a bit implausible and pretentious, so adds: "Except when I do my laundry."